1,758 research outputs found
On the Singular Neumann Problem in Linear Elasticity
The Neumann problem of linear elasticity is singular with a kernel formed by
the rigid motions of the body. There are several tricks that are commonly used
to obtain a non-singular linear system. However, they often cause reduced
accuracy or lead to poor convergence of the iterative solvers. In this paper,
different well-posed formulations of the problem are studied through
discretization by the finite element method, and preconditioning strategies
based on operator preconditioning are discussed. For each formulation we derive
preconditioners that are independent of the discretization parameter.
Preconditioners that are robust with respect to the first Lam\'e constant are
constructed for the pure displacement formulations, while a preconditioner that
is robust in both Lam\'e constants is constructed for the mixed formulation. It
is shown that, for convergence in the first Sobolev norm, it is crucial to
respect the orthogonality constraint derived from the continuous problem. Based
on this observation a modification to the conjugate gradient method is proposed
that achieves optimal error convergence of the computed solution
Three real-space discretization techniques in electronic structure calculations
A characteristic feature of the state-of-the-art of real-space methods in
electronic structure calculations is the diversity of the techniques used in
the discretization of the relevant partial differential equations. In this
context, the main approaches include finite-difference methods, various types
of finite-elements and wavelets. This paper reports on the results of several
code development projects that approach problems related to the electronic
structure using these three different discretization methods. We review the
ideas behind these methods, give examples of their applications, and discuss
their similarities and differences.Comment: 39 pages, 10 figures, accepted to a special issue of "physica status
solidi (b) - basic solid state physics" devoted to the CECAM workshop "State
of the art developments and perspectives of real-space electronic structure
techniques in condensed matter and molecular physics". v2: Minor stylistic
and typographical changes, partly inspired by referee comment
Quasiconvex Programming
We define quasiconvex programming, a form of generalized linear programming
in which one seeks the point minimizing the pointwise maximum of a collection
of quasiconvex functions. We survey algorithms for solving quasiconvex programs
either numerically or via generalizations of the dual simplex method from
linear programming, and describe varied applications of this geometric
optimization technique in meshing, scientific computation, information
visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure
Stable low-rank matrix recovery via null space properties
The problem of recovering a matrix of low rank from an incomplete and
possibly noisy set of linear measurements arises in a number of areas. In order
to derive rigorous recovery results, the measurement map is usually modeled
probabilistically. We derive sufficient conditions on the minimal amount of
measurements ensuring recovery via convex optimization. We establish our
results via certain properties of the null space of the measurement map. In the
setting where the measurements are realized as Frobenius inner products with
independent standard Gaussian random matrices we show that
measurements are enough to uniformly and stably recover an
matrix of rank at most . We then significantly generalize this result by
only requiring independent mean-zero, variance one entries with four finite
moments at the cost of replacing by some universal constant. We also study
the case of recovering Hermitian rank- matrices from measurement matrices
proportional to rank-one projectors. For rank-one projective
measurements onto independent standard Gaussian vectors, we show that nuclear
norm minimization uniformly and stably reconstructs Hermitian rank- matrices
with high probability. Next, we partially de-randomize this by establishing an
analogous statement for projectors onto independent elements of a complex
projective 4-designs at the cost of a slightly higher sampling rate . Moreover, if the Hermitian matrix to be recovered is known to be
positive semidefinite, then we show that the nuclear norm minimization approach
may be replaced by minimizing the -norm of the residual subject to the
positive semidefinite constraint. Then no estimate of the noise level is
required a priori. We discuss applications in quantum physics and the phase
retrieval problem.Comment: 26 page
HARES: an efficient method for first-principles electronic structure calculations of complex systems
We discuss our new implementation of the Real-space Electronic Structure
method for studying the atomic and electronic structure of infinite periodic as
well as finite systems, based on density functional theory. This improved
version which we call HARES (for High-performance-fortran Adaptive grid
Real-space Electronic Structure) aims at making the method widely applicable
and efficient, using high performance Fortran on parallel architectures. The
scaling of various parts of a HARES calculation is analyzed and compared to
that of plane-wave based methods. The new developments that lead to enhanced
performance, and their parallel implementation, are presented in detail. We
illustrate the application of HARES to the study of elemental crystalline
solids, molecules and complex crystalline materials, such as blue bronze and
zeolites.Comment: 17 two-column pages, including 9 figures, 5 tables. To appear in
Computer Physics Communications. Several minor revisions based on feedbac
A new method for aspherical surface fitting with large-volume datasets
In the framework of form characterization of aspherical surfaces, European National Metrology Institutes (NMIs) have been developing ultra-high precision machines having the ability to measure aspherical lenses with an uncertainty of few tens of nanometers. The fitting of the acquired aspherical datasets onto their corresponding theoretical model should be achieved at the same level of precision. In this article, three fitting algorithms are investigated: the Limited memory-Broyden-Fletcher-Goldfarb-Shanno (L-BFGS), the Levenberg–Marquardt (LM) and one variant of the Iterative Closest Point (ICP). They are assessed based on their capacities to converge relatively fast to achieve a nanometric level of accuracy, to manage a large volume of data and to be robust to the position of the data with respect to the model. Nev-ertheless, the algorithms are first evaluated on simulated datasets and their performances are studied. The comparison of these algorithms is extended on measured datasets of an aspherical lens. The results validate the newly used method for the fitting of aspherical surfaces and reveal that it is well adapted, faster and less complex than the LM or ICP methods.EMR
- …